Editing Gamma function (section)

=== Calculating products ===
The gamma function's ability to generalize factorial products immediately leads to applications in many areas of mathematics; in [[combinatorics]], and by extension in areas such as [[probability theory]] and the calculation of [[power series]]. Many expressions involving products of successive integers can be written as some combination of factorials, the most important example perhaps being that of the [[binomial coefficient]].  For example, for any complex numbers {{mvar|z}} and {{mvar|n}}, with {{math|{{abs|''z''}} < 1}}, we can write
<math display="block">(1 + z)^n = \sum_{k=0}^\infty \frac{\Gamma(n+1)}{k!\Gamma(n-k+1)} z^k,</math>
which closely resembles the binomial coefficient when {{mvar|n}} is a non-negative integer,
<math display="block">(1 + z)^n = \sum_{k=0}^n \frac{n!}{k!(n-k)!} z^k = \sum_{k=0}^n \binom{n}{k} z^k.</math>

The example of binomial coefficients motivates why the properties of the gamma function when extended to negative numbers are natural. A binomial coefficient gives the number of ways to choose {{mvar|k}} elements from a set of {{mvar|n}} elements; if {{math|''k'' > ''n''}}, there are of course no ways. If {{math|''k'' > ''n''}}, {{math|(''n'' − ''k'')!}} is the factorial of a negative integer and hence infinite if we use the gamma function definition of factorials—dividing by infinity gives the expected value of 0.

We can replace the factorial by a gamma function to extend any such formula to the complex numbers. Generally, this works for any product wherein each factor is a [[rational function]] of the index variable, by factoring the rational function into linear expressions. If {{math|''P''}} and {{math|''Q''}} are monic polynomials of degree {{mvar|m}} and {{mvar|n}} with respective roots {{math|''p''{{sub|1}}, …, ''p{{sub|m}}''}} and {{math|''q''{{sub|1}}, …, ''q{{sub|n}}''}}, we have
<math display="block">\prod_{i=a}^b \frac{P(i)}{Q(i)} = \left( \prod_{j=1}^m \frac{\Gamma(b-p_j+1)}{\Gamma(a-p_j)} \right) \left( \prod_{k=1}^n \frac{\Gamma(a-q_k)}{\Gamma(b-q_k+1)} \right).</math>

If we have a way to calculate the gamma function numerically, it is very simple to calculate numerical values of such products. The number of gamma functions in the right-hand side depends only on the degree of the polynomials, so it does not matter whether {{math|''b'' − ''a''}} equals 5 or 10<sup>5</sup>. By taking the appropriate limits, the equation can also be made to hold even when the left-hand product contains zeros or poles.

By taking limits, certain rational products with infinitely many factors can be evaluated in terms of the gamma function as well. Due to the [[Weierstrass factorization theorem]], analytic functions can be written as infinite products, and these can sometimes be represented as finite products or quotients of the gamma function. We have already seen one striking example: the reflection formula essentially represents the sine function as the product of two gamma functions. Starting from this formula, the exponential function as well as all the trigonometric and hyperbolic functions can be expressed in terms of the gamma function.

More functions yet, including the [[hypergeometric function]] and special cases thereof, can be represented by means of complex [[contour integral]]s of products and quotients of the gamma function, called [[Barnes integral|Mellin–Barnes integral]]s.